Complexity and approximability of the happy set problem

Abstract In this paper we study the approximability of the Maximum Happy Set problem ( MaxHS ) and the computational complexity of MaxHS on graph classes: For an undirected graph G = ( V , E ) and a subset S ⊆ V of vertices, a vertex v is happy if v and all its neighbors are in S; otherwise unhappy. Given an undirected graph G = ( V , E ) and an integer k, the goal of MaxHS is to find a subset S ⊆ V of k vertices such that the number of happy vertices is maximized. MaxHS is known to be NP-hard. In this paper, we design a ( 2 Δ + 1 ) -approximation algorithm for MaxHS on graphs with maximum degree Δ. Next, we show that the approximation ratio can be improved to Δ if the maximum degree Δ of the input graph is a constant. Then, we show that MaxHS can be solved in polynomial time if the input graph is restricted to block graphs, or interval graphs. We prove nevertheless that MaxHS on bipartite graphs or on cubic graphs remains NP-hard.